Atoms in four-element generating sets of partition lattices

TL;DR

This paper proves that all elements of height one or two in the partition lattice Part(n), including atoms, are contained in four-element generating sets, and provides a simple proof that Part(∞) is four-generated as a complete lattice.

Contribution

It establishes that elements of height one or two in Part(n) belong to four-element generating sets and offers a concise proof for the four-generation of the infinite partition lattice.

Findings

Atoms and elements of height two are in four-element generating sets.

Provides a simple proof for the four-generation of Part(∞).

Connects lattice generation with cryptography and recent constructions.

Abstract

Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove that each element of Part($n$) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper "Generating Boolean lattices by few elements and exchanging session keys", see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part($n$). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, "Four-element generating sets with block count width at most two in partition lattices", available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part($n$).